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Properties

Label 2-735-15.8-c1-0-14
Degree $2$
Conductor $735$
Sign $-0.329 + 0.944i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 − 1.85i)2-s + (−0.170 − 1.72i)3-s + 4.85i·4-s + (−2.22 + 0.265i)5-s + (−2.87 + 3.50i)6-s + (5.29 − 5.29i)8-s + (−2.94 + 0.588i)9-s + (4.60 + 3.62i)10-s − 2.17i·11-s + (8.37 − 0.829i)12-s + (2.87 + 2.87i)13-s + (0.835 + 3.78i)15-s − 9.89·16-s + (4.03 + 4.03i)17-s + (6.53 + 4.35i)18-s + 3.45i·19-s + ⋯
L(s)  = 1  + (−1.30 − 1.30i)2-s + (−0.0985 − 0.995i)3-s + 2.42i·4-s + (−0.992 + 0.118i)5-s + (−1.17 + 1.43i)6-s + (1.87 − 1.87i)8-s + (−0.980 + 0.196i)9-s + (1.45 + 1.14i)10-s − 0.657i·11-s + (2.41 − 0.239i)12-s + (0.797 + 0.797i)13-s + (0.215 + 0.976i)15-s − 2.47·16-s + (0.979 + 0.979i)17-s + (1.54 + 1.02i)18-s + 0.792i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.329 + 0.944i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321733 - 0.453181i\)
\(L(\frac12)\) \(\approx\) \(0.321733 - 0.453181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.170 + 1.72i)T \)
5 \( 1 + (2.22 - 0.265i)T \)
7 \( 1 \)
good2 \( 1 + (1.85 + 1.85i)T + 2iT^{2} \)
11 \( 1 + 2.17iT - 11T^{2} \)
13 \( 1 + (-2.87 - 2.87i)T + 13iT^{2} \)
17 \( 1 + (-4.03 - 4.03i)T + 17iT^{2} \)
19 \( 1 - 3.45iT - 19T^{2} \)
23 \( 1 + (-1.26 + 1.26i)T - 23iT^{2} \)
29 \( 1 - 1.00T + 29T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 + (-6.85 + 6.85i)T - 37iT^{2} \)
41 \( 1 + 3.38iT - 41T^{2} \)
43 \( 1 + (1.85 + 1.85i)T + 43iT^{2} \)
47 \( 1 + (-1.02 - 1.02i)T + 47iT^{2} \)
53 \( 1 + (-0.260 + 0.260i)T - 53iT^{2} \)
59 \( 1 + 8.25T + 59T^{2} \)
61 \( 1 - 5.61T + 61T^{2} \)
67 \( 1 + (-6.54 + 6.54i)T - 67iT^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (1.31 + 1.31i)T + 73iT^{2} \)
79 \( 1 - 6.89iT - 79T^{2} \)
83 \( 1 + (2.04 - 2.04i)T - 83iT^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (-12.4 + 12.4i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35911725194935482490339700857, −9.114085184954200358600440724487, −8.429847548216810198244925704131, −7.897469411387226249291352781876, −7.10100076004562605097678672055, −5.92605451809714738311170966540, −3.93656503854638235867459468614, −3.23831697355137081441342432264, −1.89052504372559980447926500236, −0.78686341599622144128281930012, 0.75294067208519589007860442218, 3.21925782717097682128880356563, 4.66283272802256018842440828575, 5.35904338827691729630109028879, 6.42582453358227570092900481763, 7.47141403578612909512165274927, 8.038383643780675780562673665019, 8.908815040838636851256837826962, 9.541743211662051190543504289626, 10.33364887834015945453793491212

Graph of the $Z$-function along the critical line